- #1

user1139

- 72

- 8

$$\int_{\mathcal{V'}}\int_{\mathcal{V}}\frac{\mathbf{\mathrm{x}}\times[\mathbf{\mathrm{J_1(\mathbf{\mathrm{x'}})}}[\mathbf{\mathrm{J_2(x)}}\cdot(\mathbf{\mathrm{x}-\mathbf{\mathrm{x'}}})]]}{|\mathbf{\mathrm{x}}-\mathbf{\mathrm{x'}}|^3}\,\mathrm{d^3}x\,\mathrm{d^3}x'$$

to

$$\int_{\mathcal{V'}}\int_{\mathcal{V}}\frac{\mathbf{\mathrm{J_2(\mathbf{\mathrm{x}})}}\times\mathbf{\mathrm{J_1(\mathbf{\mathrm{x'}})}}}{|\mathbf{\mathrm{x}}-\mathbf{\mathrm{x'}}|}\,\mathrm{d^3}x\,\mathrm{d^3}x'$$

I was stuck after rewriting the first expression as

$$-\int_{\mathcal{V'}}\int_{\mathcal{V}}[\mathbf{\mathrm{x}}\times\mathbf{\mathrm{J_1(\mathbf{\mathrm{x'}})}}]

\left[\mathbf{\mathrm{J_2(x)}\cdot\nabla\left(\frac{1}{|\mathbf{\mathrm{x}-\mathbf{\mathrm{x'}}}|}\right)} \right] \,\mathrm{d^3}x\,\mathrm{d^3}x'$$